Counting with combinations
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- So let's keep thinking about different ways to sit multiple people in the certain number of chairs. So let's say we have six people. We have person A, we have person B, we have person C, person D, person E, and we have person F. So we have six people. And for the sake of this video, we're going to say oh we want to figure out all the scenarios, all the possibilities, all the permutations, all the ways that we could put them into three chairs. So there's chair number one, chair number two and chair number three. This is all review. This is covered in the permutations video. But it'll be very instructive as we move into a new concept. So what are all the permutations of putting six different people into three chairs? Well, like we've seen before, we can start with the first chair. And we can say look if no one's sat-- If we haven't seated anyone yet, how many different people could we put in chair number one? Well there's six different people who could be in chair number one. Let me get a different color. There are six people who could be in chair number one. Six different scenarios for who sits in chair number one. Now for each of those six scenarios, how many people, how many different people could sit in chair number two? Well each of those six scenarios we've taken one of the six people to sit in chair number one. So that means you have five out of the six people left to sit in chair number two. Or another way to think about it is there's six scenarios of someone in chair number one and for each of those six, you have five scenarios for who's in chair number two. So you have a total of 30 scenarios where you have seated six people in the first two chairs. And now if you want to say well what about for the three chairs? Well for each of these 30 scenarios, how many different people could you put in chair number three? Well you're still going to have four people standing up not in chairs. So for each of these 30 scenarios, you have four people who you could put in chair number three. So your total number of scenarios, or your total number of permutations where we care who's sitting in which chair is six times five times four, which is equal to 120 permutations. Permutations. Now lemme, permutations. Now it's worth thinking about what permutations are counting. Now remember we care, when we're talking about permutations, we care about who's sitting in which chair. So for example, for example this is one permutation. And this would be counted as another permutation. And this would be counted as another permutation. This would be counted as another permutation. So notice these are all the same three people, but we're putting them in different chairs. And this counted that. That's counted in this 120. I could keep going. We could have that, or we could have that. So our thinking in the permutation world. We would count all of these. Or we would count this as six different permutations. These are going towards this 120. And of course we have other permutations where we would involve other people. Where we have, it could be F, B, C, F, C, B, F, A, C, F, F. Actually let me do it this way. I'm going to be a little bit more systematic. F, uh lemme do it. B, B, F, C, B, C, F. And obviously I could keep doing. I can do 120 of these. I'll do two more. You could have C, F, B. And then you could have C, B, F. So in the permutation world, these are, these are literally 12 of the 120 permutations. But what if we, what if all we cared about is the three people we're choosing to sit down, but we don't care in what order that they're sitting, or in which chair they're sitting. So in that world, these would all be one. This is all the same set of three people if we don't care which chair they're sitting in. This would also be the same set of three people. And so this question. If I have six people sitting in three chairs, how many ways can I choose three people out of the six where I don't care which chair they sit on? And I encourage you to pause the video, and try to think of what that number would actually be. Well a big clue was when we essentially wrote all of the permutations where we've picked a group of three people. We see that there's six ways of arranging the three people. When you pick a certain group of three people, that turned into six permutations. And so if all you want to do is care about well how many different ways are there to choose three from the six? You would take your whole permutations. You would take your number of permutations. You would take your number of permutations. And then you would divide it by the number of ways to arrange three people. Number of ways to arrange, arrange three people. And we see that you can arrange three people, or even three letters. You can arrange it in six different ways. So this would be equal to 120 divided by six, or this would be equal to 20. So there are 120 permutations here. If you said how many different arrangements are there of taking six people and putting them into three chairs? That's 120. But now we're asking another thing. We're saying if we start with 120 people, and we want to choose. Sorry if we're starting with six people and we want to figure out how many ways, how many combinations, how many ways are there for us to choose three of them? Then we end up with 20 combinations. Combinations of people. This right over here, once again, this right over here is just one combination. It's the combination, A, B, C. I don't care what order they sit in. I have chosen them. I have chosen these three of the six. This is a combination of people. I don't care about the order. This right over here is another combination. It is F, C, and B. Once again I don't care about the order. I just care that I've chosen these three people. So how many ways are there to choose three people out of six? It is 20. It's the total number of permutations. It's 120 divided by the number of ways you can arrange three people.